The effect of coupling in automotive active suspensions

by

KAMAL MOUSSA MLEK

S.B., MASSACHUSETTS INSTITUTE OF TECHNOLOGY (1982)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE

DEGREE OF MASTER OF SCIENCE

IN

MECHANICAL ENGINEERING

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

May 1985

1 Kamal Moussa Malek 1985

Signature redacted

Certified b

Accepted by

f Author

Department of Mechanical Engineering May 24, 1985

Signature redacted

Signature redacted

Archives

APR 28 1986

-by

Kamal Moussa Malek

Submitted to the Department of Mechanical Engineering

on May 24, 1985 in partial fulfillment of the

requirements for the Degree of Master of Science in

Mechanical Engineering

ABSTRACT

In this study, a 7 degree-of-freedom vehicle model is

used to analyze how the coupling, which is present between

the various vehicle modes, affects the ride quality and

roadholding performance when sprung mass absolute body

velocity feedback is used in the control strategy. This

full-vehicle model makes it possible to study the response

in one mode when the vehicle is subjected to an input in

another mode.

_{It also allows the study of the tire normal}

load distribution between the four corners.

Beside providing absolute body velocity feedback, the

active system is also used to make the suspension fully

interconnected.

_{This allows the independent specification}

of all the suspension rates (stiffnesses, absolute and

relative damping rates), as well as the coupling rates.

Frequency response results obtained using existing

vehicle data show that when absolute damping is used,

decoupling the heave and pitch sprung mass modes both

improves body isolation, and reduces fore-aft tire load

transfer as the vehicle traverses an uneven road.

This

indicates that "flat-ride" turning may not be desirable when

absolute body damping is used.

_{Absolute body roll damping,}

while improving body isolation, increases the effect of road

disturbances on lateral load transfer distribution.

_{It is}

also found that for a road warp input, increased relative

damping is superior to absolute body damping.

Thesis Supervisor:

_{Dr. J. Karl Hedrick}

I would like to dedicate this modest work to my

parents, M. and M. Their love and unfailing support over the years, and the sense of purpose which they have

instilled in me ultimately remain the driving force behind any endeavour.

My deepest gratitude goes to Big Brother Aziz who can be always counted on for his willingness to listen and for his good advice, as well as his boundless help. I cannot thank him enough.

I also want to thank my sister Ilham and brother Adel for their support and help.

I am indebted to Professor J. Karl Hedrick for being interested in supervising this thesis.

Many thanks to Terri Demeris for the excellent job she did in typing this manuscript with minimal turnover time. Her ability to dispose in an nour of what took days to generate was sobering, to say the least.

I would also like to thank all my friends at the

Vehicle Dynamics Lab, in particular Ademola, Dan, Eduardo, Pat, and Sean.

Finally, this page would not be complete without a

posthumus mention of the late "lemon" (yellow) 914 which got me interested in vehicle dynamics in the first place. It lived by Trailing-Throttle Oversteer, and died by it.

MTITLE PAGE - - - -- - - - - -- - - - -ABSTR ACKNO TABLE LIST LIST CHAPT 1.1 1.2

1.3

The Passive Suspension... The Active Suspension ...

Objectives of the Study ... Scope of the Study ...

'ER 2 - VEHICLE MODEL... Introduction ...

Overview of Vehicle Models Model Development ...

2.3.1 Model Description ... 2.3.2 Coordinate Transformation 2.3.3 Equation Derivation

'ER 3 - VEHICLE CONFIGURATIONS Symmetry and Coupling ... 3.1.1 Symmetry ... 3.1.2 Heave-Pitch Coupling 3.1.3 Roll-Warp Coupling... .L o 4

1.5

1.6

2.1

2.2

2.3

3.1

i

ii

iii

iv

vi

2

3

6

10

13

18

21

32

32

33

36

37

39

45

59

59

59

62

64

3.2 Control Strategy ... 3.3 Implementation ...

3.4 Vehicle Configurations

...

4.2.1 Ride Quality Performance ... 4.2.2 Roadholding Performance ... CHAPTER 5 - CONCLUSION ...

5.1 Conclusions ...

5.2 Recommendations for Further Investigation

REFERENCES ... ... APPENDIX A ... Pac e

66

70

72

76

76

79

80

87

111

111

113

115

118

Table

No. Title Paqe

4.1 Vehicle Parameters 93

4.2 Damped Natural Frequencies (Hz) and Damping

Ratios of the Various Vehicle Modes 94 4.3 Damped Natural Frequencies, Damping Ratios, 95

and Eigenvector Amplitudes for the Heave-Pitch Subsystem

Figure

No. Title Paqe

1.1 A Typical Vehicle Suspension Model 22

1.2 Limits of Vertical Vibration, as 23 Recommended by Janeway

1.3 Fatigue or Decreased Proficiency Limits 24 Due to Whole-Body Vibration in the

Vertical Direction, as Recommended by the ISO

1.4 The Effect of Normal Tire Load (Fz) on 24 the Available Tractive Effort (Fx) and

Lateral Force at Various Slip Angles

1.5 Lateral Cornering Force as a Function of 25 Normal Tire Load, for Various Sideslip

Angles

1.6 Simple Suspension Model (De Carbon), With 26 Typical Values Shown on the Side

1.7 Transmissibility of Road Input to the 27 Sprung Mass, for the System Shown in

Fig. 1.6

1.8 Variation in Tire Normal Load, Due to 28 Road Input, for the System Shown in

Fig. 1.6

1.9 Simple Active Suspension Model 29

1.10 Schematic Diagram of One of Four 30 Controllers Used in the Westinghouse

Inertially Stabilized Automobile

1.11 Simple Active Suspension Model, With 31 Absolute Body Velocity Feedback

2.1 Various Vehicle Models 54

2.2 Seven d.o.f. Vehicle Model Used in 57 Present Study

Coordinate Transformation

Title

Vertical Acceleration at Driver's Location, Under Heave Input

Sprung Mass Heave Response to Heave Sprung Mass Pitch Response to Heave Figure No. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 Heave Due Pitch Due Paa e 96

Input

Input

to

to

Warp Due to

to Warp Input Warp Due to

Roll Due to

Fore/Aft Tire Load Transfer Due to Heave Input

Fore/Aft Tire Load Transfer Due to Pitch Input

Tire Lateral Load Transfer Difference (LLTD) Between Front and Rear Axles, Due to Roll Input

Tire Lateral Load Transfer Difference (LLTD) Between Front and Rear Axles, Due to Warp Input

97 98 99 100 101 102

103

105

106

107

109

110

Sprung Mass Roll Response Suspension Compression in Input

Suspension Compression in Warp Input

Sprung Mass Roll Response Suspension Compression in Warp Input

Suspension Compression in Warp Input

INTRODUCTION

1.1 BACKGROUND

Of all the numerous inventions since the industrial

revolution, the automobile or motor vehicle has undoubtedly

had one of the greater impacts on society in general.

Recent studies have indicated that despite the energy

shortages of the last decade and the accelerating reduction

in the world oil reserves, and despite the ever-tightening

emission regulations against both noise and atmospheric

pollution, the automobile is not likely to be replaced soon

by alternate modes of transportation.

In fact, it is

expected that the automobile, in one form or another, will

remain the main mode of personal transportation well into

the next century.

Over the one hundred years of its existence, the motor

vehicle, as an engineering system, has been gaining in

sophistication and complexity at an exponential pace.

This

has been due to various reasons, including advances in

technical knowledge and analytical capability on the part of

the designer, an increased demand from the consumer for a

better product, and fierce competition among the

manufac-turers to satisfy (or create) that demand.

Needless to say,

the microprocessor has been playing a major role in making

the car a more sophisticated system.

This trend is expected

seen today in the automobile market, and the resulting

need to provide a constantly improved product. Because the suspension, as a major subsystem of the automobile, directly influences many of its qualities and performance aspects, both subjective and objective, (ride quality, noise and vibration, handling, cornering, active safety, etc...), it has received increasing attention from the automotive engineer, and will continue to do so.

1.2 THE AUTOMOBILE SUSPENSION

The suspension is "the system of devices (such as springs) supporting the upper part of a vehicle on the

axles" (Webster's Seventh New Collegiate Dictionary). From that definition it is clear that the term "suspension" is a misnomer, and that "support" would be a better one. The term has been carried over from the days of the horse-drawn carriage, when the body was actually suspended from flexible

leaf springs which rose over their attachment points to be body.

The "upper part of a vehicle" in the above definition refers to the body or sprung mass; this consists of the structure housing the passenger and cargo compartments, as well as their contents, the engine and gearbox, and other mechanical parts. The "axles" refer to the unsprung masses of the vehicle; these consist of the tires and wheels, the

outboard brakes, and part of the driveshafts and of the

suspension and steering linkages.

Depending on the

parti-cular suspension design, there may be a solid axle, or there

may only be an imaginary line as in the case of an

inde-pendent suspension system.

In the case of a live rear axle,

the rear unsprung mass includes the differential unit and

its housing.

Strictly speaking, the unsprung masses are not

unsprung since the tires have finite stiffnesses.

Figure

1.1 schematically shows a vehicle and its suspension

system.

The suspension system has to perform two distinct

functions.

The first is to support and isolate the sprung

mass (mainly the occupants and the cargo) from any external

disturbances, such as road irregularities and wind gusts.

The second is to control the motion of the unsprung masses

so as to keep the tires in firm contact with the road

despite any disturbing inputs.

1.2.1

Ride Quality

The effectiveness of the suspension system in

perform-ing the first

function, namely isolating the body,

deter-mines the ride quality and comfort of the vehicle.

Although

vibrations in the vehicle can be caused by the various

sub-systems of the vehicle, for instance the drivetrain, or by

aerodynamic disturbances, those induced by road surface

by the vehicle occupants.

As with anything that depends

on

human perception, it

is impossible to objectively determine

precise measures for ride quality and comfort limits.

This

is caused by the variations in sensitivity from one

individ-ual.to another, and by the different methods used by the

various investigators who have undertaken such studies.

Nevertheless, there exists a large body of knowledge dealing

with the response of humans to vibration.

Of the many ride comfort criteria which have been

pro-posed, the best known are the Janeway comfort criteria

pub-lished by the Society of Automotive Engineers [1], and the

International Standard ISO 2631, 1974 [2], which deals with

whole body vibration.

Janeway divided the sinusoidal

fre-quency range from 1 to 60 Hz into three ranges, and

suggest-ed acceptable vertical vibration amplitude limits for each

range.

From 1 to 6 Hz, maximum values for the jerk (or rate

of change of acceleration) amplitude are given; from 6 to 20

Hz, peak values of vertical velocity amplitude are given;

and from 20 to 60 Hz, limits are given for the peak vertical

velocity amplitude.

These limits are shown in Figure 1.2.

The ISO standards deal only with

RMS

acceleration

resulting from sinusoidal vibration over the 1 to 80 Hz

frequency range; however, they not only consider vertical

vibration, but also longitudinal and

lateral ones.

They

and acceleration amplitude. These guidelines are the

"exposure limits" beyond which safety would be impaired, the "fatigue or decreased proficiency boundaries", and the

"reduced confort" boundaries. Figure 1.3 shows the limits for whole-body vibration for fatigue or decreased profi-ciency in the vertical direction.

Although the ISO standards are stricter for lateral and longitudinal accelerations, experiments conducted by the Motor Industry Research Association (MIRA) of England in

1960 and 1964 found very good correlation between ride comfort as perceived by the occupants, and vertical accel-eration. More specifically, in the 7-50 Hz range, the vertical acceleration of the floor correlated with the comfort ratings given by the test subject, whereas in the

0.75 - 6 Hz the vertical acceleration of the seat showed better correlation. Pitch and roll were not found to play any significant role in the ride quality. For these

reasons, vertical acceleration of the passenger is generally considered to be a good measure of ride comfort.

It should be noted that although the Janeway criteria and the ISO standards only consider vibrations above 1 Hz, the range from 0.5 to 1 Hz is also important, as it covers the frequencies at which motion sickness is felt when large amplitudes are present.

the fact that even if two persons are equally sensitive to

vibrations, the two might have different notions of what

constitutes good ride.

Where one might prefer a softly

sprung well-isolated luxury car, the other might find a

harder ride, which allows more road inputs to filter

through, more desirable, as it provides a greater feeling

of control.

1.2.2 Handling

Vehicle handling is an all-encompassing term which

reflects objective measures of the vehicle performance, as

well as the subjective "feel" it conveys to the driver

during maneuvers.

The maneuvers in question may involve

acceleration or braking, turning, or any combination

there-of.

_{The cornering ability of the vehicle is one aspect of}

handling which can be measured objectively, using a circular

skidpad.

There are other tests used to measure the

road-holding capacity

and steering response of the vehicle, for

instance the slalom course, the J-turn, and others, but

these rely more on the skill and sensitivity of the driver.

Other more subjective aspects of handling include the

feed-back received by the driver through the steering wheel

(on-center feel, cornering force feedback), the brake pedal

feel, and more.

Because of all these parameters, it is not

possible to trace the handling qualities of a vehicle to one

handling.

For the purpose of this study, the aspect of handling

that is of interest is roadholding, which can be considered

the ability of the suspension system to successfully

maintain the tires in firm contact with the road.

The need to maintain the contact force at the tire/road

interface stems from the fact that the forces needed to

guide and control the vehicle are generated at that

inter-face.

These forces include the tractive effort needed for

braking and accelerating, and the lateral tire forces needed

for maintaining a given course despite external

disturb-ances, or for changing direction.

The capacity of a

pneumatic tire to generate a longitudinal, lateral, or

combined force is a nonlinear function of the normal load

acting on it.

Figure 1.4 presents data corresponding to a

typical pneumatic tire.

It shows families of curves, each

family corresponding to a different tire normal load FZe

In

each family, the x-axis corresponds to the longitudinal

force or tractive effort FX, while the y-axis corresponds to

the lateral or cornering force Fy.

Each curve, which is a

quarter of what is known as the tire ellipse, corresponds to

a constant angle of sideslip.

The angle of sideslip is

defined as that between the wheel plance and the actual

direction of travel of the wheel.

For a given slip angle,

of generating.

It should be noted that as the normal load

increases, both forces increase for a given slip angle.

This trend levels off beyond a certain point.

All the above

applies when there is no gross relative motion between the

tire and the road.

At higher levels of slip, either

longi-tudinal or lateral (sideslip), for instance when the tire is

locked, its force-generating potential is considerably

reduced.

From the above, it is clear that a good suspension

system should attempt to reduce as much as possible any

variation in the tire contact-force that results from road

irregularities.

That would make the vehicle handle better

over rough roads, other things (such as the amount of toe

and camber variation with suspension compression) being

invariant.

However, it

is not enough for the suspension to

main-tain the total tire contact-force at the four corners of the

vehicle constant.

It is more important to maintain the

desired distribution of the contact-forces between the 4

tires,

despite any disturbing input.

It is this

distribu-tion that will determine the basic handling characteristics

of a vehicle, other parameters being equal (weight

distri-bution, relative tire sizes, inflation pressures, suspension

geometry, wheel alignment, and many others)and that will

vehicle.

Before further explanation, a few terms will be

defined. The terms understeer and oversteer have been the subject of several definitions but a simple one will be used here. Understeer refers to the condition during

steady-state cornering where the front tires are experienc-ing larger sideslip angles than the rear ones. Oversteer refers to the opposite situation [2]. In general, oversteer is considered undesirable, as it makes the vehicle unstable above a certain critical speed, when subjected to lateral

disturbances [2,4].

Whether a vehicle will understeer or oversteer during steady-state cornering is mainly determined by the relation-ship between the lateral transfer of tire normal load at the

front axle and that at the rear axle. This is due to the fact that for a given angle of sideslip, the lateral force generated by the tire is a highly nonlinear function of the

tire normal load, as shown in Figure 1.5. The total

lateral force generated by the axle is the sum of the two forces generated by each tire, but because of the decreasing slope of the curves in Figure 1.5, two equally loaded

tires will generate more lateral force than two tires with the same total load divided unequally between them.

There-fore, if the tire load transfer is greater at the front than at the rear in an otherwise symmetric (front to rear)

front to generate the same lateral force as the rear, and

the vehicle will understeer.

The vehicle will oversteer if

the opposite is true.

This is explained in more details in

reference [5], from which Figure 1.5 is taken.

During a combined turning and braking, or turning and

accelerating maneuver, the understeer or oversteer

charac-teristics of the vehicle are affected by the relative

dis-tribution of braking forces between the front and rear

axles, as well as by the front-to-rear tire load transfer.

Generally, the suspension, steering, and braking

systems are designed to provide specific handling

character-istics when traversing smooth roads.

Ideally, these

charac-teristics should be maintained over rough roads.

The level

to which the handling characteristics are unaffected by road

roughness is a measure of the success of a given suspension

design.

_{For instance a very stiffly sprung sports car that}

handles well over smooth roads, but is so unsettled by

bumps as to become difficult to manage cannot be considered

successful.

1.3

THE PASSIVE SUSPENSION

One of the simpler models used to study automotive

sus-pensions, known as the De Carbon model, is shown in Figure

1.6.

It is a linear, two degree-of-freedom

(d.o.f.),

lumped

element model, with motion in the vertical or heave

direc-first

representing the sprung mass or body, and the second

representing the unsprung or axle masses (the nomenclature

used follows standard SAE nomeclature [6] whenever

possi-ble).

The tire is modeled as a simple spring with constant

KT equal to the tire rate.

Tire damping, due to hysteresis

effects in the tire itself, is neglected.

The supporting

media are the spring K and the damper D.

The road

distur-bances are modeled as a displacement input to the tire.

This simple ride model can be used to gain some qualitative

feel as to how a suspension works, and what the compromises

involved are.

The data given in Figure 1.6 are typical of a

high-performance luxury GT car.

It can be seen that the sprung

mass is an order of magnitude larger than the unsprung one,

whereas the suspension spring rate is smaller than the tire

rate by more than an order of magnitude.

Because of that,

the frequency of the mode of vibration associated with

motion of the sprung mass is separated by more than a decade

from the frequency of the mode consisting mainly of unsprung

mass vibration.

The first

is about 1 Hz whereas the second

is above 20 Hz.

The suspension system thus acts as a

low-pass filter, as far as the sprung mass is concerned.

Figure

1.7 is a frequency response plot of the sprung mass motion

amplitude per unit input amplitude, over the frequency range

from 0.1 Hz to 100 Hz, for the system shown in Figure 1.6.

ratios of 0.25 and 0.5 respectively.

It can be seen that

for input frequencies below that of the body resonance, the

body follows the input.

The response peaks at the body

resonant frequency, after which it decreases steadily,

except for another local peak at the frequency corresponding

to the unsprung mass resonance.

The body is effectively

isolated from high frequency inputs.

One of the compromises facing the suspension designer

is selecting the optimum damping.

From Figure 1.7 it

can be

seen that using the higher damping ratio reduces the motion

transmissibility at the low frequency unsprung mass

reson-ance, and would thus provide better passenger isolation

around that frequency.

The trade-off however occurs at

higher frequencies, where the increased damping results in

worsening isolation up to the high frequency resonant peak

which is not affected much.

Beyond that peak, the roll-off

rate is decreased also.

This is due to the fact that the

higher frequencies result in an ever-increasing relative

velocity across the damper which consequently "hardens".

As

mentioned above, the second peak is not reduced by the

increased damping, some tire damping would be needed to

achieve such an improvement.

The increased damping is also beneficial as far as tire

normal load is concerned.

_{Figure 1.8 shows the variation in}

that load from the static equilibrium value, as a function

results in smaller variation at the two resonant peaks, as

well as more uniform variation between these two

frequen-cies.

In order to circumvent the low-frequency versus

high-frequency isolation, as well as other compromises of

the passive suspension, the active suspension system was

proposed.

1.4 THE ACTIVE SUSPENSION

In the previous section, we saw one aspect of the

limitations of passive suspensions in isolating the body

from road disturbances.

That limitation was due to the

reliance on a simple damper in parallel with the spring as

supporting media.

Since the 1930's, suspension systems

which would eliminate the shortcomings of that arrangement

have been considered.

Significant work on these active

suspension systems has been carried out since the 1950's,

following advances in the development of high-performance

servo-mechanisms using feedback control.

In their 1975

review of the

state-of-the-art of active

suspensions for ground vehicles

[7],

Hedrick and Wormley

define an active suspension as one that may rely on

externally-supplied power for its operation.

In a typical

hydraulic, pneumatic, electric, or any combination.

Unlike the passive suspension which, by virtue of its

spring and dampers can only temporarily store, return, or

dissipate the energy input due to road disturbances, the

actuator force (FA) in the active suspension does not depend

upon the input time-history.

The energy input to the active

suspension can be modulated according to any arbitrary

control law.

This frees the suspension from the limitations

imposed by the constitutive relations of the

spring and the

damper.

The suspension force is no longer function of the

relative displacement and the relative velocity only, but it

can be function of any variable, either local or remotely

measured.

Also, these measured variables can be arbitrarily

processed before being used to generate the suspension

force.

The active suspension can thus be made to adapt to

varying inputs, road conditions, or just driver taste.

Because of its need for an external power source, the

active suspension suffers from disadvantages relating to

cost, complexity, and safety in the case of a power failure.

Moreover, as with any complex

system, maintenance becomes an

important requirement, especially in view of the rough

operating environment involved.

The heart of an active suspension system is the control

strategy used to generate the suspension force.

Many

reflect factors such as body isolation, suspension stroke requirement (rattlespace), actuator force or power require-ments. Again, the state-of-the-art review by Hedrick and Wormley [7], or a later one by Goodall and Kortum [8], cover the various developments of active suspension systems.

One of the first applications of active suspension in the automotive field is the system designed and built by the Westinghouse Research Laboratories in the lates 1950's [9].

The control strategy used is based on the sprung mass accel-eration, mostly in the vertical direction, but also in the horizontal plane. The system relied on acceleration feed-back to increase the effective sprung-mass inertia several-fold, thus reducing the sprung-mass resonance frequency and improving ride quality. Additional passive damping is used to control the increased displacement at the reduced fre-quency. The controllers, installed at each corner of the sprung-mass, consist of a spring-damper mounted masses with feedback compensation, and rather complicated fluidic

circuits (Fig. 1.10). These controllers can be tilted in the longitudinal or frontal plane to provide inertial stabi-lization against pitch and roll, or they can be turned

around to cause banking of the sprung mass during corner-ing.

design an active suspension system, and considers two

different arrangements, both based on a

2 d.o.f.,

heave-only model.

One arrangement consists of the actuator

mounted parallel to the spring, whereas in the other both

are mounted in series.

A sensor arrangement which allows

banking control of the body is also presented.

A system which may appear similar to the

above-mentioned Westinghouse design, but which is actually quite

different, is that developed by Automotive Products of

England, and tested in a Rover automobile [11].

This system

is a high-gain self-levelling system which counteracts body

motions due to dynamic body forces generated during

cornering, and acceleration or braking.

It does not respond

to road disturbance inputs, which are left for the passive

component of the suspension to deal with.

Another control strategy often used is full state

variable feedback.

It has been used both in the context of

pole placement techniques [12,13,14], as well as optimal

control [15,16].

Hullender et al. [15]

show that for a one

d.o.f. suspension system subjected to a random input, the

optimal suspension consists of a spring supporting a mass,

and of an absolute damper, i.e. an active force actuator

responding to absolute body velocity feedback (Fig. 1.11).

That suspension is optimal with respect to a quadratic

performance index penalizing suspension stroke as well as

combined with body acceleration feedback, to improve the vertical and lateral ride quality, as well as the lateral stability of railway cars.

Absolute body velocity feedback, or absolute damping has received increased attention lately [14,17,18], and is the control strategy of interest in the present work. One of the attractions of absolute damping is its relative ease of implementation, as only one sensor is needed to measure body acceleration, and a single integration of that

signal yields the absolute body velocity. Another advantage lies in the possibility of extending the results obtained with an active absolute damper to a semi-active one, with

little loss of system performance, using a modulable passive damper [14]. A semi-active system is one which does not

require external power input, but in which the passive forces are modulated according to some control strategy.

In Ref. [17], the concept of absolute velocity feedback is extended to the unsprung mass in a 2 d.o.f. heave-only model in an attempt to control wheel hop. It is shown that even an active suspension system still presents a trade-off as far as providing body isolation while at the same time controlling the motion of the unsprung masses. This

conclusion, similar to the one Thompson reaches in [10], is caused by a limitation due to the structural layout of the

and opposite forces to the sprung and unsprung masses.

Karnopp [18]

reaches similar conclusions, but also

develops a simple criterion to help determine whether active

damping will bring any significant performance improvements

in a given application.

1.5 OBJECTIVES OF THE STUDY

Most studies reviewed in the previous section utilize

Heave-only,

1

or 2 d.o.f. models to design or to analyze the

performance of active suspensions. Because of their rela-tive simplicity, these models are very useful for under-standing active suspensions and for designing control strategies, as demonstrated by the vast body of knowledge that has been built around them. In designing a full

suspension system, one could use a one-dimensional two-mass model for each of the vehicle modes, namely heave, pitch, roll and warp. Such an approach however would not reflect the fact that there is coupling between those modes. Static coupling between heave and pitch is usually unavoidable

because it may be necessary to fit springs of unequal stiffness at the front and at the rear, even in a vehicle with equal front-to-rear weight distribution. This may be done for various reasons:

9 vehicles without load-levelling devices may require stiffer springs under the cargo compartment, in

under full load;

* the designer may choose specific locations for the

pitch and bounce centers of the sprung mass, in

order to obtain a given ride characteristic, a "flat

ride" at a given speed for instance [2,19];

* the choice may be made in order to obtain

the desired handling characteristics.

Stiffer

springs at the rear could be used to alter the

front-to-rear suspension roll couple distribution

and cause more tire load transfer at the rear,

resulting in less understeer (or more oversteer)

[5,20].

The suspension roll and warp modes are usually coupled

because of the desirability of having unequal roll couples

at the front and at the rear.

As the above indicates, coupling between the various

modes is a very important tool used by the designer to

confer on the vehicle specific handling and ride

characteristics, and therefore cannot be neglected.

Although both Thompson [10], and Hanna and Osbon [9]

consider a full

vehicle with banking control, they only

provide an analysis of the steady-state banking moment

needed.

The effect that active suspension control has on

the handling characteristics as the vehicle traverses an

uneven road is not discussed.

heave and pitch motions, but the model he uses does not include unsprung masses, and therefore cannot predict the effect of the control strategy on tire normal loads, and thus on handling.

From the point of view of ride comfort, Margolis found that active damping results in decreased body isolation in certain cases. Specifically, the body pitch response to a heave input, and vice versa, increased compared to the

passive system. This effect was more marked in the case of full state variable feedback.

The objective of the present study is to analyze active suspension systems using a model which reflects the coupling present between the various degrees of freedom in a real vehicle. Such a model should also allow the effect of active control on vehicle handling to be assessed.

In a first step, a model including 7 d.o.f. is

developed. In that model the remote sensing capability of the active suspension is used to provide full interconnec-tion between the four corners of the vehicle, and thus to develop the most general suspension system, while maintain-ing the symmetry requirement. An example of suspension

interconnection are the anti-roll bars, or stabilizers which are common in today's cars. Another is the Moulton

front-to-rear interconnected suspension [21].

effect of coupling on vehicle ride quality and handling,

when body absolute velocity feedback is used.

That control

strategy was chosen for the advantages which have been

described previously.

A partially decoupled suspension configuration is used

to eliminate the disadvantages found by Margolis [14], while

at the same time leaving the handling characteristics

unaffected.

1.6

SCOPE OF THE STUDY

This first

chapter has presented some necessary

back-ground material concerning suspension systems in general, as

well as some ride confort and handling considerations.

Active suspensions were presented, along with a review

of previous work of particular interest to the present

study.

The vehicle model used, along with the corresponding

equations of motion, are presented in Chapter 2.

Chapter 3 discusses the various suspension

configura-tions, including coupling and absolute damping.

Results corresponding to the various configurations,

and based on parameters from a current production

automo-bile, are presented in Chapter 4.

Finally, the conclusions of the study and

recommenda-tions for future work are summarized in Chapter 5.

OPITCH SPRUNG BOUNCE MASS z SUSPENSION SPRING SHOCK ABSORBER UNSPRUNG MASS

TIRE SPRING TIRE DAMPING

Fig. 1.1 A Typical Vehicle Suspension Model (From Ref. [2])

mm in 10 8 6 RECOMMENDED LIMITS 100 - 4 -2 f=1TO6Hz MAX 10 JERK 8 CONSTANT 6 10 - 4 f =6 TO 20Hz 2 MAX ACCELERATION 0.10 CONSTANT 8 6 4-f =20 TO 60 Hz MAX 2 ~VELOCITY CONSTANT 0.01 -< 6 01 4 2 0.001 8 6 0.01 - 4 2 0.0001 1 2 4 6 8 10 20 40 60 Hz VIBRATION FREOUENCY f

Fig. 1.2 Limits of Vertical Vibration, as Recommended by Janeway

10.0 Er1.0 x 9

50-z

-005

Fig. 1.3 Fatigue or Decreased Proficiency Limits Due To Whole-Body Vibration in the Vertical Direction, as Recommended by the ISO (From Ref. [2])

F

z-

z

F

8*

Fig. 1.4 The Effect of Normal Tire Load (Fz) on the Available Tractive Effort (Fx) and Lateral Force at Various Slip Angles

(From Ref. [3])

G670-15 - 24 PSI INFLATION 2'r

r-.

Fig. 1.5 Lateral Cornering Force as a Function of Normal Tire Load, for Various Sideslip Angles (From Ref. [5])

F

M

k

body=

Fig. 1.6 Simple Suspension Model (De Carbon), With Typical Values Shown on the Side

SIMPLE

SUSPENSION

-- 25% CRITICAL BODY DAMPING --- 50% CRITICAL BODY DAMPING

>-U

0

10

FREQUENCY

(HZ)

Fig. 1.7 Transmissibility of Road Input to the Sprung Mass, for the System Shown in Fig. 1.6 1 10 0 10 10 1 -2 10 -3 -4 _L 10 . I 100 ' ' ' I'

SIMPLE SUSPENSION

1 10

FREQUENCY

(HZJ

Fig. 1.8 Variation in Tire Normal Load, Due to Road Input, for the System Shown in Fig. 1.6 5 10

z

z

-- 25% CRITICAL BODY DAMPING 50% CRITICAL BODY DAMPING

21

FA: active control force

-Z _-Z

Simple Active Suspension Model MF

FA

rn

Accelerotion

Cover

Sensing Memer _{Cover Vent} Compensating Piston

I LeofDapn

oht Feedback I Spring - Damping Dashpot

Piston I Pivot

c'G

Valve _LReservoir

Prse _{Main Supply} Anti-sing _{Fixed Pilot}

Orif ice _Orifice

Exhaust Valve _{H req. Min. Press.}

Bias Filter-Co. _Valve

Pressure

Exhaust Inlet Valve Damping et P _{Orifice and} Floating N N Valves

Plug Reservoir Actuator Actuator Upper Lower

Cylinder Cylinder

Fig. 1.10 Schematic Diagram of One of Four Controllers Used in the Westinghouse Inertially Stabilized Automobile

Z _B

z B

SF _A

Fig. 1.11 Simple Active Suspension Model, With Absolute Body Velocity

Feedback

FA : active control force

VEHICLE MODEL

2.1 INTRODUCTION

The automobile is a rather complex engineering system. From a system dynamics point of view, an accurate model representing all the subsystems that can vibrate would

involve multiple degrees of freedom. To start with, such a model would include the dynamic modes of the vehicle body. Beside the 6 d.o.f.'s needed to describe the gross motions of the vehicle, an infinite number would be required to

model the vibratory modes of the body structure itself, such as torsion and bending. The sub-assemblies such as the

engine and transmission, the propeller shaft (if present) and the driveshafts, the various components of the suspen-sion, the steering system (including the steering column and steering wheel),, as well as others, should be included.

These various asemblies are not rigidly mounted to the

vehicle structure, and can thus vibrate relative to it. In fact, such vibrations are very often the cause of occupant discomfort in actual automobiles, and are the subject of

extensive analysis by the manufacturers. Unfortunately, each of these assemblies would require several d.o.f.'s for a complete description. A complete dynamic model of the

vehicle should include the rotating and reciprocating

torques that result from these rotating inertias can affect handling, not to mention the vibrations they generate.

Moreover, the occupants are suported by the seats which are relatively compliant, and they should be treated as sub-assemblies of the whole system, even if the human body is assumed a rigid mass instead of the mass-spring-damper

system that it is. Clearly, such an accurate model would be difficult to generate and even more difficult to analyze. Besides requiring sizeable computational power, its

complexity will result in the loss of any physical insight into the system being analyzed.

As a general rule, the best model is the simplest one which can represent the phenomena being analyzed, and

provide the answers sought. For instance, a more complex model would be needed if accurate numerical results, as opposed to general trends, are needed.

2.2 OVERVIEW OF VEHICLE MODELS

In general, vehicle models tend to fall into one of two categories: ride models, and handling models. In the higher order models, this distinction is no longer valid however.

The simplest ride model is the one-mass, heave-only model used, for example, by Hullender et. al. [15]. Because of the large difference in bandwidth between the sprung and

One step above in complexity

is the two-mass, heave-only model, which includes both

sprung and unsprung masses.

This model, which was described

in Chapter 1, has been widely used [2,10,13,17,18].

Another

two d.o.f. model is the heave-pitch model described by Wong

[2], and used by Margolis [14].

This model is closer to the

one-mass heave model, since it does not include the unsprung

masses.

_{However, by treating the body as a non-symmetric}

beam supported by two unequal springs, it accounts for the

coupling present between the heave and the pitch degrees of

freedom of the vehicle.

A model combining the

characteris-tics of the last two models mentioned can be used to analyze

the unsprung mass vibration at the same time as body heave

and pitch.

_{A seven d.o.f. model, involving three for the}

body, and one for each of the four unsprung masses is

des-cribed by Wong [2].

It allows the study of vehicle response

to inputs with different profiles under the right and the

left tracks.

_{Such a model is used by Mercier [22], but only}

in the case of a vehicle which is symmetric with respect to

a longitudinal and a lateral vertical planes.

The next step

of refinement, as far as ride models are concerned, would be

to include the seat suspension and the mass of the occupant.

Figure 2.1 shows the various models mentioned above. It

is worth noting that all these models treat the vehicle as

a linear lumped-parameter system, and they all assume the

given road profile.

The main difference between ride models and handling

models, is that the formers do not account for the lateral

displacement of the vehicle, or for its yaw or rotation

around the heave axis, they thus assume that the road

traversed is straight.

The simplest handling model, known

as the bicycle model [2,3], consists of two degrees of

freedom, one for vehicle sideslip, and one for vehicle yaw.

The unsprung masses are not modeled separately, and the

vehicle is not allowed to roll; the tires are modeled as

lateral stiffnesses.

The model permits investigating the

effect of the location of the center of gravity, and the

front and rear tire stiffnesses, on the understeer or

over-steer characteristics of the vehicle during constant speed

maneuvers.

A more refined handling model is the one

pro-posed by Segel [23], and used widely.

The model considers

the vehicle

as a three-dimensional system, consisting of an

unsprung mass representing the whole vehicle, with freedom

to yaw and sideslip, as well as a sprung mass constrained to

roll about an axis fixed to the unsprung mass.

Another

degree of freedom could be added for the steering system, to

account for the inertia and compliance there.

In the latter

case, it becomes possible to study the "free control" of the

vehicle, where a torque input is applied to the steering

wheel, as opposed to "fixed control" where the steering

A handling model using a totally different approach is

the one developed by Ellis [24]. In this model, no roll axis

is assumed, and the unsprung masses are not given separate

degrees of freedom, but they are assumed "geared" to the

sprung mass.

Therefore, they cannot vibrate independently,

and the model is only valid over smooth roads with no

disturbances, although degrees of freedom for body heave and

body pitch are provided.

The six degrees of freedom are

those of the whole vehicle.

Variable forward speed, as well

as coupling effects due to the particular suspension

geometry are accounted for.

Higher order models, which can be used to analyze ride

quality and handling at the same time, have been developed,

and are commonly used.

The simplest has ten degrees of

freedom, six for the body and one for each of the four

unsprung masses.

Eleven degrees of freedom can be used, if

the steering system dynamics are of interest.

Some models

include a rotational degree of freedom for each of the

unsprung masses, normal to the wheel plane.

An example of

such a model is that used at the University of Michigan

[25].

2.3

MODEL DEVELOPMENT

The model used in this study is a seven d.o.f. ride

model.

The model is first described below, and its

advant-description of a coordinate transformation which is of importance in deriving the equations of motion and in analyzing the system. Finally, in a third section, the equation derivation is outlined.

2.3.1 Model Description

The model used is shown schematically in Figure 2.2. It has seven degrees of freedom:

* Three degrees of freedom for the sprung mass which is assumed rigid. Relative to a reference origina-ting from the center of gravity of the sprung mass, these are the heave H, or vertical translation, the roll R, or rotation about the longitudinal axis, and pitch P, or rotation about the lateral axis, as

shown in Fig. 2.2;

* One degree of freedom for each unsprung mass. The four unsprung masses are lumped independently. Each is assumed to move in a direction normal to the

plane of the body. Furthermore, the unsprung masses can only transmit forces through the suspension

springs, dampers, and active elements. These forces are assumed normal to the plane of the sprung mass. It should be noted that the coordinates chosen are only valid for small displacements and rotations, since the

were used to locate the body in space. The rotations have to be small enough for the small angle approximation to be valid.

The whole system is assumed linear, and the stiff-nesses, both of the suspension and of the tires, as well as the damping rates, whether relative or absolute, are con-stant. Although a real vehicle exhibits many non-lineari-ties, both geometric and constitutive, the linear model is justifiable since the object of the study is not to obtain accurate numerical results, but rather to understand the general behavior of the system.

The small amount of damping present in the tires is neglected, and the tires are thus modeled as linear springs. In terms of energy dissipation, tire damping has a secondary effect compared to the damping present in the suspension

[2]. In fact, if the body is assumed inertially fixed, and the unsprung masses are allowed to vibrate, tire hysteresis would provide only about 10% of critical damping for the unsprung masses, according to results obtainied by Overton, Mills, and Ashley [26].

Another assumption concerning the tires is that they remain in contact with the road at all time. The road

itself is assumed to provide the only inputs to the system, as the vehicle travels at a constant speed in a straight line.

respect to a longitudinal vertical plane through its middle.

Although the model described above is a ride model, it is adequate for understanding the effect of road distur-bances on the handling characteristics of the vehicle,

because it is a full three-dimensional model. It is possible to keep track of all four tire normal loads individually, and thus to analyze how their distribution varies as the

vehicle traverses a bumpy road. The extent to which the normal load distribution is affected by the disturbances, can be used to qualitatively compare various suspension designs, or control strategies. This approach presents

certain limitations however, when quantitative estimates are desired: a given change in normal load distribution could have no appreciable effect at low vehicle acceleration, yet result in a dramatic change in handling at higher accelera-tions, because the effect of normal load on the available tire forces becomes more pronounced closer to the tire limit of adhesion (ref. Fig. 1.5). Because of this and other

non-linear behavior, simple superposition does not apply.

2.3.2 Coordinate Transformation

It is desirable to obtain the equations of motion in a form that lends itself to simple physical interpretation. A crucial step in this direction involves the transformation of all the variables and parameters at the four corners into

pitch, and roll).

This can be accomplished via linear

com-binations of these "local" variables, as described in this

sub-section.

It should be noted that this transformation,

as well as the equation derivation which follows in the next

sub-section, are loosely based on those of Mercier [22],

although definitions may differ, partly because the vehicle

is not assumed symmetric, and for other reasons as well.

The transformation will be first

derived for the road

inputs.

The four corners of the vehicle are numbered from 1 to

4, starting at the left front corner and moving clockwise.

Referring to Figure 2.3, the problem could be stated as

follows:

Given the road vertical displacements ZRi, i

=

1 to 4,

under each wheel, taken positive downward, a transformation

matrix [TF] is needed, such that:

{QR} = [TF] {ZR};

where {QR} is the new road input vector consisting of road

heave HR, road pitch PR, road roll RR, and road warp WR;

HR

{QR}= PR

RR

WR

new variables are defined. This is facilitated by the introduction of the mid-point displacements ZR1 2, ZR3 4,

ZR_{1 4}, and ZR_{2 3} shown in Figure 2.3, where ZRij is the

weighted average vertical displacement of the road plane at the points corresponding to the intersections of the x- and y-axes with the vehicle perimeter:

ZRy

ZR2

ZR

1

2

(1)

and

ZR

b

(4)

where a is the distance from the center of gravity of the

sprung mass, or from its projection onto the road plane, to

the front axle, and b is the distance from the same point to

the rear axle.

The heave of the road plane is now defined as the

vertical distance between the projection of the sprung mass

center-of-gravity onto the static or zero-input road plane,

or upon substituting for ZR1 2 and ZR3 4 from Eqns. (1) and

(2):

b - ZR₁ + b - ZR₂ + a - ZR₃ + a * ZR₄

HR = 2(a + b) _. (6)

The pitch of the road plane is defined as the angle formed by the projections of the x-axis on the static, and on the displaced road planes:

-ZR2

ZR

Similarly, road roll is defined as the angle between projections of the y-axis onto a static and a displaced road planes:

ZR2

- ZR

RR = ZR23 -Z14 ₂ ,(9) tm

where tm is the half track of the vehicle, measured along the y-axis. _{From Fig. 2.3, it can be seen that:}

b-

tf

_tr

axle)

,

and tr is the rear half track.

RR =

-b - ZR + b - ZR2 + a - ZR3 - a - ZR₄

2(b - tf + a - tr) (11) after _{substituting for tm, and for ZR1 4 and ZR2 3}

-Finally, road plane warp is defined as the conjugate to road roll. It gives an indication of the torsional deflec-tion of the road plane:

-b ZR + b * ZR2 - a - ZR₃ + a * ZR₄ 2(b - tf + a * tr)

(12)

above definitions, [TF] can be obtained:

b

a

a

b

a

-a

2q

2q

2q

2w

2w

a

-l

-l

where w is the wheelbase:

w = a + b

q

b

tf

(13)

and

define tire and suspension compressions and unsprung mass displacements.

The tire compression terms ZTi, i = 1 to 4, corres-ponding to the four corners, can be combined into tire heave HT, tire pitch PT, tire roll RT, and tire warp WT:

HT _ZT 1 PT= [TF] ZT₂ (16)

RT

ZT

Similarly for suspension compression terms ZSi:

HS

_PS~

= [TF] * , (17)

RS ZS3

and for the unsprung mass displacements:

HU ZU1

PU = [TF] * ZU_{2 (18)}

RU _ZU

3

.W ; ZU4j

The advantage of using the transformed coordinates and variables will become apparent later on in the study. The transformations make the system more transparent as far as analyzing and understanding the coupling between the various modes, and its effect on the response of the system.